one positive and one negative (which is of course consistent with (REF ) being strongly indefinite).
For each positive \(\sigma \) , of which there are at most finitely many, there are two eigenvalues \(\rho =\rho _\pm \) 
(a double eigenvalue for \(\sigma =-\frac{1}{2}\) ), both with positive real part.
In particular, all eigenvalues of (REF ) lie in the union \(\lbrace  \text{Im}(z)=0 \rbrace  \cup \lbrace  \text{Re}(z)=\frac{c}{2} \rbrace  \subset \mathbb {C}\) .
Hence, when parameters are varied eigenvalues can only pass from the left half-plane to the right half-plane through the origin.
It is thus reasonable to expect that
for \(c>0\)  (or \(c<0\) ) the spectral flow for the linearization of (REF ) is well-defined, and this is indeed the case, see [1]}, [2]}. Furthermore, it follows from (REF ) that along a homotopy eigenvalues \(\rho \)  of (REF ) and \(\sigma \)  of (REF ) cross the origin simultaneously and in the same direction.
The spectral flows for (REF ) and (REF ) are thus “the same” in the sense that the relative index of a pair of equilibria for (REF ) is equal to the relative index of this pair for (REF ). Since the latter is easier to analyse (it is scalar), we compute relative indices using the parabolic equation (REF ) and then draw conclusions for the strongly indefinite system (REF ), or, equivalently, the travelling wave problem (REF ).
